Under the state of equilibrium, in statics, is understood the state in which all parts of the mechanical system are at rest relative to some inertial coordinate system. One of the basic objects of statics are forces and points of their application.
The force acting on a material point with a radius vector from other points is a measure of the influence of other points on the considered point, as a result of which it receives acceleration relative to the inertial reference frame. Value strength is determined by the formula:
,
where m is the mass of the point - a value that depends on the properties of the point itself. This formula is called Newton's second law.
Application of statics in dynamics
An important feature of the equations of motion of an absolutely rigid body is that forces can be converted into equivalent systems. With such a transformation, the equations of motion retain their form, but the system of forces acting on the body can be transformed into a simpler system. Thus, the point of application of force can be moved along the line of its action; forces can be expanded according to the parallelogram rule; forces applied at one point can be replaced by their geometric sum.
An example of such transformations is gravity. It acts on all points of a rigid body. But the law of motion of the body will not change if the force of gravity distributed over all points is replaced by a single vector applied at the center of mass of the body.
It turns out that if we add an equivalent system to the main system of forces acting on the body, in which the directions of the forces are reversed, then the body, under the action of these systems, will be in equilibrium. Thus, the task of determining equivalent systems of forces is reduced to the problem of equilibrium, that is, to the problem of statics.
The main task of statics is the establishment of laws for the transformation of a system of forces into equivalent systems. Thus, the methods of statics are used not only in the study of bodies in equilibrium, but also in the dynamics of a rigid body, in the transformation of forces into simpler equivalent systems.
Material point statics
Consider a material point that is in equilibrium. And let n forces act on it, k = 1, 2, ..., n.
If the material point is in equilibrium, then the vector sum of the forces acting on it is equal to zero:
(1)
.
In equilibrium, the geometric sum of the forces acting on a point is zero.
Geometric interpretation. If the beginning of the second vector is placed at the end of the first vector, and the beginning of the third is placed at the end of the second vector, and then this process is continued, then the end of the last, nth vector will be combined with the beginning of the first vector. That is, we get a closed geometric figure, the lengths of the sides of which are equal to the modules of the vectors. If all vectors lie in the same plane, then we get a closed polygon.
It is often convenient to choose rectangular coordinate system Oxyz. Then the sums of the projections of all force vectors on the coordinate axes are equal to zero:
If we choose any direction defined by some vector , then the sum of the projections of the force vectors on this direction is equal to zero:
.
We multiply equation (1) scalarly by the vector:
.
Here is the scalar product of the vectors and .
Note that the projection of a vector onto the direction of the vector is determined by the formula:
.
Rigid body statics
Moment of force about a point
Determining the moment of force
Moment of force, applied to the body at point A, relative to the fixed center O, is called a vector equal to the vector product of the vectors and:(2) .
Geometric interpretation
The moment of force is equal to the product of the force F and the arm OH.
Let the vectors and be located in the plane of the figure. According to the property of the cross product, the vector is perpendicular to the vectors and , that is, perpendicular to the plane of the figure. Its direction is determined by the right screw rule. In the figure, the moment vector is directed towards us. The absolute value of the moment:
.
Since , then
(3)
.
Using geometry, one can give another interpretation of the moment of force. To do this, draw a straight line AH through the force vector . From the center O we drop the perpendicular OH to this line. The length of this perpendicular is called shoulder of strength. Then
(4)
.
Since , formulas (3) and (4) are equivalent.
Thus, absolute value of the moment of force relative to the center O is product of force on the shoulder this force relative to the chosen center O .
When calculating moment, it is often convenient to decompose the force into two components:
,
where . The force passes through the point O. Therefore, its momentum is zero. Then
.
The absolute value of the moment:
.
Moment components in rectangular coordinates
If we choose a rectangular coordinate system Oxyz centered at the point O, then the moment of force will have the following components:
(5.1)
;
(5.2)
;
(5.3)
.
Here are the coordinates of point A in the selected coordinate system:
.
The components are the values of the moment of force about the axes, respectively.
Properties of the moment of force about the center
The moment about the center O, from the force passing through this center, is equal to zero.
If the point of application of force is moved along a line passing through the force vector, then the moment, during such a movement, will not change.
The moment from the vector sum of forces applied to one point of the body is equal to the vector sum of the moments from each of the forces applied to the same point:
.
The same applies to forces whose extension lines intersect at one point.
If the vector sum of the forces is zero:
,
then the sum of the moments from these forces does not depend on the position of the center, relative to which the moments are calculated:
.
Power couple
Power couple- these are two forces, equal in absolute value and having opposite directions, applied to different points of the body.
A pair of forces is characterized by the moment they create. Since the vector sum of the forces included in the pair is zero, the moment created by the couple does not depend on the point relative to which the moment is calculated. From the point of view of static equilibrium, the nature of the forces in the pair is irrelevant. A pair of forces is used to indicate that a moment of forces acts on the body, having a certain value.
Moment of force about a given axis
Often there are cases when we do not need to know all the components of the moment of force about a selected point, but only need to know the moment of force about a selected axis.
The moment of force about the axis passing through the point O is the projection of the vector of the moment of force, about the point O, on the direction of the axis.
Properties of the moment of force about an axis
The moment about the axis from the force passing through this axis is equal to zero.
The moment about an axis from a force parallel to this axis is zero.
Calculation of the moment of force about an axis
Let a force act on the body at point A. Let us find the moment of this force relative to the O′O′′ axis.
Let's build a rectangular coordinate system. Let the Oz axis coincide with O′O′′ . From the point A we drop the perpendicular OH to O′O′′ . Through the points O and A we draw the axis Ox. We draw the axis Oy perpendicular to Ox and Oz. We decompose the force into components along the axes of the coordinate system:
.
The force crosses the O′O′′ axis. Therefore, its momentum is zero. The force is parallel to the O′O′′ axis. Therefore, its moment is also zero. By formula (5.3) we find:
.
Note that the component is directed tangentially to the circle whose center is the point O . The direction of the vector is determined by the right screw rule.
Equilibrium conditions for a rigid body
In equilibrium, the vector sum of all forces acting on the body is equal to zero and the vector sum of the moments of these forces relative to an arbitrary fixed center is equal to zero:
(6.1)
;
(6.2)
.
We emphasize that the center O , relative to which the moments of forces are calculated, can be chosen arbitrarily. Point O can either belong to the body or be outside it. Usually the center O is chosen to make the calculations easier.
The equilibrium conditions can be formulated in another way.
In equilibrium, the sum of the projections of forces on any direction given by an arbitrary vector is equal to zero:
.
The sum of moments of forces about an arbitrary axis O′O′′ is also equal to zero:
.
Sometimes these conditions are more convenient. There are times when, by choosing axes, calculations can be made simpler.
Center of gravity of the body
Consider one of the most important forces - gravity. Here, the forces are not applied at certain points of the body, but are continuously distributed over its volume. For each part of the body with an infinitesimal volume ∆V, the gravitational force acts. Here ρ is the density of the substance of the body, is the acceleration of free fall.
Let be the mass of an infinitely small part of the body. And let the point A k defines the position of this section. Let us find the quantities related to the force of gravity, which are included in the equilibrium equations (6).
Let's find the sum of gravity forces formed by all parts of the body:
,
where is the mass of the body. Thus, the sum of the gravity forces of individual infinitesimal parts of the body can be replaced by one gravity vector of the entire body:
.
Let's find the sum of the moments of the forces of gravity, relative to the chosen center O in an arbitrary way:
.
Here we have introduced point C which is called center of gravity body. The position of the center of gravity, in a coordinate system centered at the point O, is determined by the formula:
(7)
.
So, when determining static equilibrium, the sum of the gravity forces of individual sections of the body can be replaced by the resultant
,
applied to the center of mass of the body C , whose position is determined by formula (7).
The position of the center of gravity for various geometric shapes can be found in the relevant reference books. If the body has an axis or plane of symmetry, then the center of gravity is located on this axis or plane. So, the centers of gravity of a sphere, circle or circle are located in the centers of the circles of these figures. The centers of gravity of a rectangular parallelepiped, rectangle or square are also located in their centers - at the points of intersection of the diagonals.
Uniformly (A) and linearly (B) distributed load.
There are also cases similar to the force of gravity, when the forces are not applied at certain points of the body, but are continuously distributed over its surface or volume. Such forces are called distributed forces or .
(Figure A). Also, as in the case of gravity, it can be replaced by the resultant force of magnitude , applied at the center of gravity of the diagram. Since the diagram in figure A is a rectangle, the center of gravity of the diagram is in its center - point C: | AC| = | CB |. (picture B). It can also be replaced by the resultant. The value of the resultant is equal to the area of the diagram:.
The point of application is in the center of gravity of the diagram. The center of gravity of a triangle, height h, is at a distance from the base. So .
Friction forces
Sliding friction. Let the body be on a flat surface. And let be a force perpendicular to the surface with which the surface acts on the body (pressure force). Then the sliding friction force is parallel to the surface and directed to the side, preventing the body from moving. Its largest value is:
,
where f is the coefficient of friction. The coefficient of friction is a dimensionless quantity.
rolling friction. Let the rounded body roll or may roll on the surface. And let be the pressure force perpendicular to the surface with which the surface acts on the body. Then on the body, at the point of contact with the surface, the moment of friction forces acts, which prevents the movement of the body. The largest value of the friction moment is:
,
where δ is the coefficient of rolling friction. It has the dimension of length.
References:
S. M. Targ, Short Course in Theoretical Mechanics, Higher School, 2010.
The course covers: kinematics of a point and a rigid body (and from different points of view it is proposed to consider the problem of orientation of a rigid body), classical problems of the dynamics of mechanical systems and the dynamics of a rigid body, elements of celestial mechanics, motion of systems of variable composition, impact theory, differential equations of analytical dynamics.
The course covers all the traditional sections of theoretical mechanics, but special attention is paid to the most meaningful and valuable for theory and applications sections of dynamics and methods of analytical mechanics; statics is studied as a section of dynamics, and in the section of kinematics, the concepts necessary for the section of dynamics and the mathematical apparatus are introduced in detail.
Informational resources
Gantmakher F.R. Lectures on Analytical Mechanics. - 3rd ed. – M.: Fizmatlit, 2001.
Zhuravlev V.F. Fundamentals of theoretical mechanics. - 2nd ed. - M.: Fizmatlit, 2001; 3rd ed. – M.: Fizmatlit, 2008.
Markeev A.P. Theoretical mechanics. - Moscow - Izhevsk: Research Center "Regular and Chaotic Dynamics", 2007.
Requirements
The course is designed for students who own the apparatus of analytical geometry and linear algebra in the scope of the first-year program of a technical university.
Course program
1. Kinematics of a point
1.1. Problems of kinematics. Cartesian coordinate system. Decomposition of a vector in an orthonormal basis. Radius vector and point coordinates. Point speed and acceleration. Trajectory of movement.
1.2. Natural triangular. Expansion of velocity and acceleration in the axes of a natural trihedron (Huygens' theorem).
1.3. Curvilinear point coordinates, examples: polar, cylindrical and spherical coordinate systems. Velocity components and projections of acceleration on the axes of a curvilinear coordinate system.
2. Methods for specifying the orientation of a rigid body
2.1. Solid. Fixed and body-bound coordinate systems.
2.2. Orthogonal rotation matrices and their properties. Euler's finite turn theorem.
2.3. Active and passive points of view on orthogonal transformation. Addition of turns.
2.4. Finite rotation angles: Euler angles and "airplane" angles. Expression of an orthogonal matrix in terms of finite rotation angles.
3. Spatial motion of a rigid body
3.1. Translational and rotational motion of a rigid body. Angular velocity and angular acceleration.
3.2. Distribution of velocities (Euler's formula) and accelerations (Rivals' formula) of points of a rigid body.
3.3. Kinematic invariants. Kinematic screw. Instant screw axle.
4. Plane-parallel motion
4.1. The concept of plane-parallel motion of the body. Angular velocity and angular acceleration in the case of plane-parallel motion. Instantaneous center of speed.
5. Complex motion of a point and a rigid body
5.1. Fixed and moving coordinate systems. Absolute, relative and figurative movement of a point.
5.2. The theorem on the addition of velocities in the case of a complex motion of a point, relative and figurative velocities of a point. The Coriolis theorem on the addition of accelerations for a complex motion of a point, relative, translational and Coriolis accelerations of a point.
5.3. Absolute, relative and portable angular velocity and angular acceleration of a body.
6. Motion of a rigid body with a fixed point (quaternion presentation)
6.1. The concept of complex and hypercomplex numbers. Algebra of quaternions. Quaternion product. Conjugate and inverse quaternion, norm and modulus.
6.2. Trigonometric representation of the unit quaternion. Quaternion method of specifying body rotation. Euler's finite turn theorem.
6.3. Relationship between quaternion components in different bases. Addition of turns. Rodrigues-Hamilton parameters.
7. Exam work
8. Basic concepts of dynamics.
8.1 Momentum, angular momentum (kinetic moment), kinetic energy.
8.2 Power of forces, work of forces, potential and total energy.
8.3 Center of mass (center of inertia) of the system. The moment of inertia of the system about the axis.
8.4 Moments of inertia about parallel axes; the Huygens–Steiner theorem.
8.5 Tensor and ellipsoid of inertia. Principal axes of inertia. Properties of axial moments of inertia.
8.6 Calculation of the angular momentum and kinetic energy of the body using the inertia tensor.
9. Basic theorems of dynamics in inertial and non-inertial frames of reference.
9.1 Theorem on the change in the momentum of the system in an inertial frame of reference. The theorem on the motion of the center of mass.
9.2 Theorem on the change in the angular momentum of the system in an inertial frame of reference.
9.3 Theorem on the change in the kinetic energy of the system in an inertial frame of reference.
9.4 Potential, gyroscopic and dissipative forces.
9.5 Basic theorems of dynamics in non-inertial frames of reference.
10. Movement of a rigid body with a fixed point by inertia.
10.1 Euler dynamic equations.
10.2 Euler case, first integrals of dynamical equations; permanent rotations.
10.3 Interpretations of Poinsot and Macculag.
10.4 Regular precession in the case of dynamic symmetry of the body.
11. Motion of a heavy rigid body with a fixed point.
11.1 General formulation of the problem of the motion of a heavy rigid body around.
fixed point. Euler dynamic equations and their first integrals.
11.2 Qualitative analysis of the motion of a rigid body in the case of Lagrange.
11.3 Forced regular precession of a dynamically symmetric rigid body.
11.4 The basic formula of gyroscopy.
11.5 The concept of the elementary theory of gyroscopes.
12. Dynamics of a point in the central field.
12.1 Binet's equation.
12.2 Orbit equation. Kepler's laws.
12.3 The scattering problem.
12.4 The problem of two bodies. Equations of motion. Area integral, energy integral, Laplace integral.
13. Dynamics of systems of variable composition.
13.1 Basic concepts and theorems on the change of basic dynamic quantities in systems of variable composition.
13.2 Movement of a material point of variable mass.
13.3 Equations of motion of a body of variable composition.
14. Theory of impulsive movements.
14.1 Basic concepts and axioms of the theory of impulsive movements.
14.2 Theorems about changing the basic dynamic quantities during impulsive motion.
14.3 Impulsive motion of a rigid body.
14.4 Collision of two rigid bodies.
14.5 Carnot's theorems.
15. Control work
Learning Outcomes
As a result of mastering the discipline, the student must:
- Know:
- basic concepts and theorems of mechanics and the methods of studying the motion of mechanical systems arising from them;
- Be able to:
- correctly formulate problems in terms of theoretical mechanics;
- develop mechanical and mathematical models that adequately reflect the main properties of the phenomena under consideration;
- apply the acquired knowledge to solve relevant specific problems;
- Own:
- skills in solving classical problems of theoretical mechanics and mathematics;
- the skills of studying the problems of mechanics and building mechanical and mathematical models that adequately describe a variety of mechanical phenomena;
- skills in the practical use of methods and principles of theoretical mechanics in solving problems: force calculation, determining the kinematic characteristics of bodies with various methods of setting motion, determining the law of motion of material bodies and mechanical systems under the action of forces;
- skills to independently master new information in the process of production and scientific activities, using modern educational and information technologies;
Point kinematics.
1. The subject of theoretical mechanics. Basic abstractions.
Theoretical mechanicsis a science in which the general laws of mechanical motion and mechanical interaction of material bodies are studied
Mechanical movementcalled the movement of a body in relation to another body, occurring in space and time.
Mechanical interaction is called such an interaction of material bodies, which changes the nature of their mechanical movement.
Statics - This is a branch of theoretical mechanics, which studies methods for converting systems of forces into equivalent systems and establishes the conditions for the equilibrium of forces applied to a solid body.
Kinematics - is the branch of theoretical mechanics that deals with the movement of material bodies in space from a geometric point of view, regardless of the forces acting on them.
Dynamics - This is a branch of mechanics that studies the movement of material bodies in space, depending on the forces acting on them.
Objects of study in theoretical mechanics:
material point,
system of material points,
Absolutely rigid body.
Absolute space and absolute time are independent of each other. Absolute space - three-dimensional, homogeneous, motionless Euclidean space. Absolute time - flows from the past to the future continuously, it is homogeneous, the same at all points in space and does not depend on the movement of matter.
2. The subject of kinematics.
Kinematics - this is a branch of mechanics that studies the geometric properties of the motion of bodies without taking into account their inertia (i.e. mass) and the forces acting on them
To determine the position of a moving body (or point) with the body in relation to which the movement of this body is being studied, rigidly, some coordinate system is connected, which together with the body forms reference system.
The main task of kinematics is to, knowing the law of motion of a given body (point), to determine all the kinematic quantities that characterize its motion (velocity and acceleration).
3. Methods for specifying the movement of a point
· natural way
Should be known:
Point movement trajectory;
Start and direction of counting;
The law of motion of a point along a given trajectory in the form (1.1)
· Coordinate method
Equations (1.2) are the equations of motion of the point M.
The equation for the trajectory of point M can be obtained by eliminating the time parameter « t » from equations (1.2)
· Vector way
|
|
(1.3) Relationship between coordinate and vector methods for specifying the movement of a point
|
(1.4)
Connection between coordinate and natural ways of specifying the movement of a point
Determine the trajectory of the point, excluding time from equations (1.2);
-- find the law of motion of a point along a trajectory (use the expression for the arc differential)
After integration, we obtain the law of motion of a point along a given trajectory:
The connection between the coordinate and vector methods of specifying the movement of a point is determined by equation (1.4)
4. Determining the speed of a point with the vector method of specifying the movement.
Let at the momenttthe position of the point is determined by the radius vector , and at the moment of timet 1
– radius-vector , then for a period of time 
the point will move.
|
|
point average speed, the direction of the vector is the same as the vector
|
(1.5)
The speed of a point at a given time
To get the speed of a point at a given moment of time, it is necessary to make a passage to the limit
(1.6)
(1.7)
The speed vector of a point at a given time is equal to the first derivative of the radius vector with respect to time and is directed tangentially to the trajectory at a given point.
(unit¾ m/s, km/h)
Mean acceleration vector has the same direction as the vectorΔ v , that is, directed towards the concavity of the trajectory.
Acceleration vector of a point at a given time is equal to the first derivative of the velocity vector or the second derivative of the point's radius vector with respect to time.
(unit - )
How is the vector located in relation to the trajectory of the point?
In rectilinear motion, the vector is directed along the straight line along which the point moves. If the trajectory of the point is a flat curve, then the acceleration vector , as well as the vector cp, lies in the plane of this curve and is directed towards its concavity. If the trajectory is not a plane curve, then the vector cp will be directed towards the concavity of the trajectory and will lie in the plane passing through the tangent to the trajectory at the pointM and a line parallel to the tangent at an adjacent pointM 1 . AT limit when the pointM 1 tends to M this plane occupies the position of the so-called contiguous plane. Therefore, in the general case, the acceleration vector lies in a contiguous plane and is directed towards the concavity of the curve.
General theorems of the dynamics of a system of bodies. Theorems on the motion of the center of mass, on the change in the momentum, on the change in the main moment of the momentum, on the change in kinetic energy. Principles of d'Alembert, and possible displacements. General equation of dynamics. Lagrange's equations.
ContentThe work done by the force, is equal to the scalar product of the force vectors and the infinitesimal displacement of the point of its application :
,
that is, the product of the modules of the vectors F and ds and the cosine of the angle between them.
The work done by the moment of force, is equal to the scalar product of the vectors of the moment and the infinitesimal angle of rotation :
.
d'Alembert principle
The essence of d'Alembert's principle is to reduce the problems of dynamics to the problems of statics. To do this, it is assumed (or it is known in advance) that the bodies of the system have certain (angular) accelerations. Next, the forces of inertia and (or) moments of inertia forces are introduced, which are equal in magnitude and reciprocal in direction to the forces and moments of forces, which, according to the laws of mechanics, would create given accelerations or angular accelerations
Consider an example. The body makes a translational motion and external forces act on it. Further, we assume that these forces create an acceleration of the center of mass of the system . According to the theorem on the movement of the center of mass, the center of mass of a body would have the same acceleration if a force acted on the body. Next, we introduce the force of inertia:
.
After that, the task of dynamics is:
.
;
.
For rotational movement proceed in a similar way. Let the body rotate around the z axis and external moments of forces M e zk act on it. We assume that these moments create an angular acceleration ε z . Next, we introduce the moment of inertia forces M И = - J z ε z . After that, the task of dynamics is:
.
Turns into a static task:
;
.
The principle of possible movements
The principle of possible displacements is used to solve problems of statics. In some problems, it gives a shorter solution than writing equilibrium equations. This is especially true for systems with connections (for example, systems of bodies connected by threads and blocks), consisting of many bodies
The principle of possible movements.
For the equilibrium of a mechanical system with ideal constraints, it is necessary and sufficient that the sum of the elementary works of all active forces acting on it for any possible displacement of the system be equal to zero.
Possible system relocation- this is a small displacement, at which the connections imposed on the system are not broken.
Perfect Connections- these are bonds that do not do work when the system is moved. More precisely, the sum of work performed by the links themselves when moving the system is zero.
General equation of dynamics (d'Alembert - Lagrange principle)
The d'Alembert-Lagrange principle is a combination of the d'Alembert principle with the principle of possible displacements. That is, when solving the problem of dynamics, we introduce the forces of inertia and reduce the problem to the problem of statics, which we solve using the principle of possible displacements.
d'Alembert-Lagrange principle.
When a mechanical system moves with ideal constraints at each moment of time, the sum of the elementary works of all applied active forces and all inertia forces on any possible displacement of the system is equal to zero:
.
This equation is called general equation of dynamics.
Lagrange equations
Generalized coordinates q 1 , q 2 , ..., q n is a set of n values that uniquely determine the position of the system.
The number of generalized coordinates n coincides with the number of degrees of freedom of the system.
Generalized speeds are the derivatives of the generalized coordinates with respect to time t.
Generalized forces Q 1 , Q 2 , ..., Q n
.
Consider a possible displacement of the system, in which the coordinate q k will receive a displacement δq k . The rest of the coordinates remain unchanged. Let δA k be the work done by external forces during such a displacement. Then
δA k = Q k δq k , or
.
If, with a possible displacement of the system, all coordinates change, then the work done by external forces during such a displacement has the form:
δA = Q 1 δq 1 + Q 2 δq 2 + ... + Q n δq n.
Then the generalized forces are partial derivatives of the displacement work:
.
For potential forces with potential Π,
.
Lagrange equations are the equations of motion of a mechanical system in generalized coordinates:
Here T is the kinetic energy. It is a function of generalized coordinates, velocities, and possibly time. Therefore, its partial derivative is also a function of generalized coordinates, velocities, and time. Next, you need to take into account that the coordinates and velocities are functions of time. Therefore, to find the total time derivative, you need to apply the rule of differentiation of a complex function:
.
References:
S. M. Targ, Short Course in Theoretical Mechanics, Higher School, 2010.
